100 guests go to a party, and some of them shake hands with each other. Show that there are two guests who handshake the same number of people.

Each of the people at the party has handshaken between 0 and 99 times. However, if someone has handshaken 0 times (with nobody), it is impossible that another one has handshaken 99 times (with everybody). Therefore there are at most 98 different options for the number of handshakes at the party, and thus two of the guests have handshaken the same number of times.